Idea

NOTE: these are messing-about experiments so the graphs below don’t have axis ticks labelled correctly.

Model

Incredibly simplified model where both rabbits and foxes live at most 2 years, and essentially reproduce when they turn 1 year old. They have fertility rates specified per individual, with no tracking of the division between males and females, and the number of babies produced by the population in total is taken as uniform variate rather than a more appropriate distribution. It also uses the oversimplified rule that if a fox needs to eat cc rabbits to survive for 1 year, then the foxes “telepathically” arrange so that an individual fox eats either exactly cc rabbits or no rabbits at all; more realistic modelling is a work-in-progress.

quantity

symbol

no of foxes born this year

f0f_0

no of 1 year old foxes

f1f_1

half of maximum offspring from pair of foxes

fff_f

no of rabbits a fox needs to eat in a year to survive

cc

no of rabbits born this year

r0r_0

no of 1 year old rabbits

r1r_1

half of maximum offspring from pair of rabbits

rfr_f

maximum carrying capacity of vegetation (in no of rabbits)

KrK_r

System equations

The populations evolve from year to year with some stochastic equations (in discrete time):

where UU denotes a fresh random variable uniformly distributed on [0,1)[0,1) for each occurrence.

Simulation results

Running the system for 2222^22 different random simulation runs for various values of fox and rabbit fertility (for fixed cc, KrK_r, etc), the “fox population has not died out” probabilities after 50 years can be visualised in the below plot (where horizontal axis left to right is increasing fox fertility and vertical axis top to bottom is increasing rabbit fertility):

The “reasonably favourable values of fertility” for foxes lie in a range of about [2,2.6][2,2.6], and providing it’s above a minimum value rabbit fertility doesn’t matter.

The surival probabilities taken as a series of slices parallel to the “rabbit fertility axis”:

At least part of the reason why the curves go to a horizontal line is that, with a fixed “rabbit carrying capacity”, above a certain rabbit fertility level it’s carrying capacity rather than rabbit fertility that determines the number of rabbits.

The surival probabilities taken as a series of slices parallel to the “fox fertility axis”:

This looks like it might be a gamma distribution (speculation: this might possibly be because the “surviving trajectories” are ones that don’t hit any of the clamping terms, so somehow in the visualised region it’s a “nice” section which is the straightforward product of uniform random variates, which may have some nice closed-form?). Running a simulation with only 2162^16 different random simulation runs has “kinks” which one couldn’t tell if are genuinely significant parts of the distribution or are “under-sampling” artifacts:

(The vertical axis tick labels are correct here: the absolute probability of surviving 50 years is indeed below 0.0060.006.)

It’s also interesting to see how the survival distribution evolves over time. Since there is an “absorbing barrier” at 0 (i.e., extinction) the survival probability can only decrease over time, so the distributions can be plotted on the same graph, with timesteps corresponding to consecutive curves moving downwards. Due to the drop off in scale making later curves difficult to see, these are plotted (with y-axis being probability, x-axis related to fox fertility) in blocks of 10 consecutive timesteps (again with a correct vertical axis labelling of probability):

Steps 0–9.

Steps 10–19.

Steps 20–29.

Steps 30–39.

As can be seen, the evolution towards the skewed beta-ish distribution after 50 steps starts with some distinctly different curve shapes.

For completeness, here are the other parameters used

Fixed system parameters

Value

cc

200

initial f0f_0

100

initial f1f_1

0

initial r0r_0

2cf02 c f_0

initial r1r_1

0

KrK_r

5cf05 c f_0

Notes

2222^22 evaluations of 50 timesteps of this incredibly simple system for a 32×3232 \times 32 grid of system parameter values took about 3 hours on my PC.